A converter like that should work fine, and with your Amiga too. Some modern TV's don't like the Amiga's non-interlaced video signal, but your CRT TV should be okay with it, especially if it's a standard 50/60Hz model (not 100Hz).

To check whether it would work, try connecting your Amiga's mono video output (phono socket) to the TV's component Y input (often a green phono socket) and see whether you get a stable monochrome image.

You might also need some kind of SCART audio breakout adaptor to get the GameCube audio going to the TV. (The Amiga would be no problem since you can connect directly to its audio outputs.)